Page 281 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
eigenfrequency ω
. These polarization oscillations couple to the electric
0
χ
field via the susceptibility , so we have
˙
∇ × H = ε E + P (7.94a)
0
∇ × E = – µ H ˙ (7.94b)
0
2
˙˙
P + ω P = χε E (7.94c)
0 0
Assume that there are transverse plane waves for the fields that propagate
in the z-direction. Suppose that the polarization vector and the electric
field vector point to the x-direction than there is a magnetic field vector
pointing in the y-direction. Thus we may do the ansatz for any of the
fields E , H , and P of the form
x y x
(
–
F = F exp ( ikz ωt)) (7.95)
0
If we insert this into the equations of motion (7.94a) this yields a charac-
teristic equation

4 2 2 2 2 2 2 2
(
ω – ω ω + χ + c k ) + ω c k = 0 (7.96)
0 0
Polariton Solving (7.96) for ω we obtain the dispersion relation for a light field
Dispersion propagating in a polarizable medium. This polarizability may be due to
Relation
optical phonons in an ionic crystal. The quantized electromagnetic-polar-
ization wave is called polariton and thus (7.96) is known as the polariton
dispersion relation. It consists of two branches divided by a frequency
region where no real solution of (7.96) can be found for arbitrary values
of k. This stop band ranges from ω = ω to ω = ω + χ . For high val-
0 0
ues of the wavevector k the upper branch behaves like a light field with
linear dispersion with the light velocity cn⁄ , while for low it is dis-
k
o
persion free as for an optical phonon. The lower branch shows inverse
behavior, it saturates for high and behaves like a light field dispersion
k
for low with a light velocity lower than c n⁄ (see Figure 7.11).
k
o
This gives us a further insight in how light propagates in a polarizable
medium. As we already saw from the discussion in Section 7.4 there is a

278 Semiconductors for Micro and Nanosystem Technology